On Energy Cascades in the Forced 3D Navier–Stokes Equations

Abstract

We show -in the framework of physical scales and (K 1 , K 2 )-averages -that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale are sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws -in terms of Grashof number -for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.2000 Mathematics Subject Classification. 35Q30, 76F02.

On the Kolmogorov dissipation law in a damped Navier-Stokes equation

“…Gr from which we can write we can write4 Gr and since by Theorem 3 we have the inequality Gr ≤ a 2,G 0 Re 2 , we can deduce from these two facts the control 2 c 0…”

“…In order to state the Kolmogorov dissipation law we need to introduce some terminology: let ε > 0 be the energy dissipation rate which determines the amount of energy lost by the viscous forces (i.e. below the Kolmogorov dissipation scale ℓ D ) in the turbulent flow and which is given as an average of the gradient of the velocity (see (4) for a precise definition). Define U > 0 as the fluid characteristic velocity which is given by an average of the velocity (see (4) below) and consider L ≥ ℓ 0 to be the fluid characteristic length which is related to the domain where we study the Kolmogorov dissipation law.…”

“…The next remark is about inequalities (3). Indeed, even if in the periodic framework we can assure that ε, U < +∞, only the upper estimate ε ≤ c 2 U 3 ℓ 0 is known under some technical conditions and the lower bound c 1 U 3 ℓ 0 ≤ ε is still an open problem (see the articles [3,4,5,6,7,21]). In the case of the whole space R 3 , the damping term and the particular characteristic length L (see ( 12) below) will play an interesting role as they will help us to prove the lower and the upper bounds:…”